One of his most famous ideas has come to be known as Russell's Teapot. Russell's Teapot was a thought experiment used to illustrate the idea that the philosophic burden of proof lies upon a person making ontologically positive claims rather than shifting the burden of proof to others, specifically in the case of religion.
By ontologically positive claims, we mean claims that something does exist, as opposed to claims that something does not exist, which cannot be proven.
In others words, if someone says, "There is a God," then it is up to them to prove it. A person cannot be expected to prove a claim such as, "There is no God."
Many orthodox people speak as though it were the business of sceptics to disprove received dogmas rather than of dogmatists to prove them. This is, of course, a mistake. If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense. If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.
The main point of the argument is that no one can prove a negative claim, and therefore Occam's razor suggests that the more simple theory (in which there is no supreme being) should trump the more complex theory (with a supreme being).
Carl Sagan used a version of this argument in his book, A Demon-Haunted World. His version was about "The Dragon in My Garage," in a chapter of the same name. It was almost an identical argument, but using an invisible dragon in place of a teapot. At the end, Sagan notes that, "Your inability to invalidate my hypothesis is not at all the same thing as proving it true."
However, this thinking is partially erroneous. Philosopher Paul Chamberlain says it is inaccurate to assert that positive truth claims bear a burden of proof while negative truth claims do not. He notes that all truth claims bear a burden of proof, and that like Mother Goose and the tooth fairy, the teapot bears the greater burden not because of its negativity but because of its triviality, arguing that "When we substitute normal, serious characters such as Plato, Nero, Winston Churchill, or George Washington in place of these fictional characters, it becomes clear that anyone denying the existence of these figures has a burden of proof equal to, or in some cases greater than, the person claiming they do exist."
In other words, examples like celestial teapots and invisible dragons (or Invisible Pink Unicorns or Flying Spaghetti Monsters) only work because they are fairly absurd to begin with. However, if I were to suggest that, "There really was no George Washington. He never existed," then the reasonable thing would be to demand some evidence, not to adopt the assertion simply because it cannot be disproved.
Even ontologically negative claims demand evidence. In some cases, the burden is even heavier on negative claims.
Earlier, I said that the reasoning behind Russell's Teapot is only "partially erroneous." That is because, typically, the burden is, in fact, heavier on the positive claim. Theists do have the burden to show some evidence for God's existence, and if there is no evidence, then they are simply making an unfounded assumption.
However, even though atheists have less of a burden, the burden is still theirs, as well, and if they can show no evidence against the existence of God, then they are simply making an unfounded assumption.
The assumption behind atheism may be a safer bet, but it is still not as logical or rational as simply not making an assumption at all. If an individual were being truly rational and logical, they would be agnostic.
A truly rational individual would not make the claim that, "There is a God," or that, "There is no God." They would not make any claim at all. They would simply say, "I don't know."